# State (functional analysis)

In functional analysis, a **state** of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both §§ Mixed states and Pure states. Density matrices in turn generalize state vectors, which only represent pure states. For *M* an operator system in a C*-algebra *A* with identity, the set of all states of* *M, sometimes denoted by S(*M*), is convex, weak-* closed in the Banach dual space *M*^{*}. Thus the set of all states of *M* with the weak-* topology forms a compact Hausdorff space, known as the **state space of M **.

In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables (self-adjoint elements of the C*-algebra) to their expected measurement outcome (real number).

## Jordan decomposition[edit]

States can be viewed as noncommutative generalizations of probability measures. By Gelfand representation, every commutative C*-algebra *A* is of the form *C*_{0}(*X*) for some locally compact Hausdorff *X*. In this case, *S*(*A*) consists of positive Radon measures on *X*, and the § pure states are the evaluation functionals on *X*.

More generally, the GNS construction shows that every state is, after choosing a suitable representation, a vector state.

A bounded linear functional on a C*-algebra *A* is said to be **self-adjoint** if it is real-valued on the self-adjoint elements of *A*. Self-adjoint functionals are noncommutative analogues of signed measures.

The Jordan decomposition in measure theory says that every signed measure can be expressed as the difference of two positive measures supported on disjoint sets. This can be extended to the noncommutative setting.

**Theorem**Every self-adjoint*f*in*A*^{*}can be written as*f*=*f*_{+}−*f*_{−}where*f*_{+}and*f*_{−}are positive functionals and ||*f*|| = ||*f*_{+}|| + ||*f*_{−}||.

**Proof**—

A proof can be sketched as follows:
Let Ω be the weak*-compact set of positive linear functionals on *A* with norm ≤ 1, and *C*(Ω) be the continuous functions on Ω.
*A* can be viewed as a closed linear subspace of *C*(Ω) (this is *Kadison's function representation*).
By Hahn–Banach, *f* extends to a *g* in *C*(Ω)* with

It follows from the above decomposition that *A** is the linear span of states.

## Some important classes of states[edit]

### Pure states[edit]

By the Krein-Milman theorem, the state space of *M* has extreme points^{[clarification needed]}. The extreme points of the state space are termed **pure states** and other states are known as **mixed states**.

### Vector states[edit]

For a Hilbert space *H* and a vector *x* in *H*, the equation ω_{x}(*A*) := ⟨*Ax*,*x*⟩ (for *A* in *B(H)* ), defines a positive linear functional on *B(H)*. Since ω_{x}(*1*)=||*x*||^{2}, ω_{x} is a state if ||*x*||=1. If *A* is a C*-subalgebra of *B(H)* and *M* an operator system in *A*, then the restriction of ω_{x} to *M* defines a positive linear functional on *M*. The states of *M* that arise in this manner, from unit vectors in *H*, are termed **vector states** of *M*.

### Normal states[edit]

A state is called **normal**, iff for every monotone, increasing net of operators with least upper bound , converges to .

### Tracial states[edit]

A **tracial state** is a state such that

For any separable C*-algebra, the set of tracial states is a Choquet simplex.

### Factorial states[edit]

A **factorial state** of a C*-algebra *A* is a state such that the commutant of the corresponding GNS representation of *A* is a factor.

## See also[edit]

## References[edit]

- Lin, H. (2001),
*An Introduction to the Classification of Amenable C*-algebras*, World Scientific